New Stick Number Bounds from Random Sampling of Confined Polygons

New Stick Number Bounds from Random Sampling of Conﬁned Polygons

Thomas D. Eddy and Clayton Shonkwiler

Department of Mathematics, Colorado State University, Fort Collins, CO

Abstract The stick number of a knot is the minimum number of segments needed to build a polygonal version of the knot. Despite its elementary definition and relevance to physical knots, the stick number is poorly understood: for most knots we only know bounds on the stick number. We adopt a Monte Carlo approach to finding better bounds, producing very large ensembles of random polygons in tight confinement to look for new examples of knots constructed from few segments. We generated a total of 220 billion random polygons, yielding either the exact stick number or an improved upper bound for more than 40% of the knots with 10 or fewer crossings for which the stick number was not previously known. We summarize the current state of the art in Appendix A, which gives the best known bounds on stick number for all knots up to 10 crossings.

1 Introduction

The stick number of a knot is the minimum number of segments needed to create a polygonal version of the knot. The equilateral stick number is defined similarly, though with the added restriction that all the segments should be the same length. Despite their elementary definitions, these invariants are only poorly understood: for example, prior to our work the exact stick number was only known for 30 of the 249 nontrivial knots up to 10 crossings. The sheer simplicity of its definition partially explains the appeal of the stick number as a knot invariant, but the stick number is also interesting as a prototypical geometric or physical knot invariant since the number of segments needed to build a knot is a measure of the geometric complexity of the knot. Polygonal knots serve as a model of ring polymers like bacterial DNA (see the excellent survey [44]) so, while this model is not particularly realistic without adding further constraints, the stick number gives an indication of the minimum size of a ring polymer (or, indeed, any knotted physical system) needed to achieve a given knot type. Any example of a polygonal knot gives an upper bound on the stick number of the knot: if we can build a polygonal version of a knot K out of 10 sticks, then the stick number of K cannot possibly be bigger than 10. Consequently, we can prove state-of-the-art bounds on stick numbers by constructing examples of knots with fewer sticks than previously observed. If we happen to construct an example of a knot K with n sticks where n is known to be a lower bound on the stick number of K, then this proves that the stick number of K is exactly n.

1 935 939 943

945 948

Figure 1: Minimal stick equilateral representations of the knots 935, 939, 943, 945, and 948. Each knot is shown in orthographic perspective, viewed from the direction of the positive z-axis.

Our approach to finding such examples is to randomly sample the space of equilateral n-stick knots for small n, which requires identifying the knot type of each sample and checking to see if it beats the previous record. Early work of Millett [36, 37] and Calvo and Millett [13] used a similar Monte Carlo approach, but the challenge is that the overwhelming majority of samples are not interesting, so getting good results requires generating a truly vast number of samples. Indeed, the best extant bounds on (equilateral) stick number are due to Scharein [49] and Rawdon and Scharein [47], who used a combination of stochastic agitation and relaxation starting from standard (rather than random) conformations. The main novelty of our Monte Carlo approach is that we generate polygonal knots with equal-length edges in very tight confinement. Since complicated knots are typically condensed, this approach is a form of enriched sampling which boosts the yield of the complicated knots we seek. Generating confined knots to explore stick number is not new in itself – Calvo and Millett [13, 37] generated stick knots with vertices uniformly distributed in the unit cube (the so-called uniform random polygon model) – but ours seems to be the first such investigation which addresses equilateral stick number. Our strategy depends, in turn, on recent advances in sampling random equilateral polygons using symplectic geometry [16, 15]. The efficiency of our algorithms combined with modern hardware leads to a substantial improvement

2 over earlier Monte Carlo experiments in the sheer size of the ensembles we produce. We generated a total of 220 billion random knots, requiring approximately 50,000 core-hours of processing time on a 64-core machine with 2.3 GHz AMD Opteron 6276 processors. The bulk of this time was devoted to identifying the knot type of each of these 220 billion knots, which we did using the HOMFLY polynomial, hyperbolic volume, and crossing number. 1 We found examples of the knots 935, 939, 943, 945, and 948 built with 9 equal-length sticks, shown in Figure 1; since the stick number of these knots is known to be at least 9, this gives the exact (equilateral) stick number for these ﬁve knots:

Theorem 1. The stick number and equilateral stick number of each of the knots 935, 939, 943, 945, and 948 is exactly 9.

As we will see, this implies that each of these knots has superbridge index 4. Also, we found new upper bounds on the equilateral stick number for 88 of the 214 knots up to 10 crossings for which it remains unknown:

Theorem 2. The equilateral stick number of each of the knots 92, 93, 911, 915, 921, 925, 927, 108, 1016, 1017, 1056, 1083, 1085, 1090, 1091, 1094, 10103, 10105, 10106, 10107, 10110, 10111, 10112, 10115, 10117, 10118, 10119, 10126, 10131, 10133, 10137, 10138, 10142, 10143, 10147, 10148, 10149, 10153, and 10164 is less than or equal to 10.

The equilateral stick number of each of the knots 103, 106, 107, 1010, 1015, 1018, 1020, 1021, 1022, 1023, 1024, 1026, 1028, 1030, 1031, 1034, 1035, 1038, 1039, 1043, 1044, 1046, 1047, 1050, 1051, 1053, 1054, 1055, 1057, 1062, 1064, 1065, 1068, 1070, 1071, 1072, 1073, 1074, 1075, 1077, 1078, 1082, 1084, 1095, 1097, 10100, and 10101 is less than or equal to 11.

The equilateral stick number of each of the knots 1076 and 1080 is less than or equal to 12. In particular, all knots up to 10 crossings have equilateral stick number ≤ 12.

An upper bound on equilateral stick number implies an upper bound on stick number, and all of the corresponding stick number bounds in Theorem 2 are new except for 10107, 10119, and 10147. These knots, together with 819, 929, 1016, and 1079, appeared on Rawdon and Scharein’s list of knots for which they could not ﬁnd an equilateral stick knot achieving a known bound on stick number, and thus were potential examples of knots for which stick number and equilateral stick number differ. Millett [38] found an example of an equilateral 8-stick 819, proving that eqstick(819) = 8, which was already known to be the stick number, and our 10-stick 1016 (see Figure 2) beats the previous best bounds on both stick number (11) and equilateral stick number (12). In turn, our observed equilateral 10-stick examples of 10107, 10119, and 10147 match the best previous bound on stick number, leaving only 929 and 1079 from Rawdon and Scharein’s list.

The new bounds on equilateral stick number for the knots 108, 1016, 1039, 1064, 1073, 10105, 10110, and 10117 in Theorem 2 are 2 smaller than the previous best bounds due to Rawdon and Scharein, and for 1084 (see Figure 2) the new bound is 3 smaller. These substantial improvements in bounds for certain knots, coupled with the fact that Rawdon and Scharein’s bounds are superior to ours for the knots 916, 101, 10109, 10113, 10114, and 10123, gives a sense of the differences between their approach and ours, which we view as complementary.

1For knots up to 10 crossings we use Alexander–Briggs notation consistent with the (Perko-corrected) Rolfsen table [48]; for knots with 11 or more crossings, we use the Dowker–Thistlethwaite naming convention.

3 1016 1084

Figure 2: The 10-stick 1016 improves the best previous bound on equilateral stick number by 2. The 11-stick 1084 improves the best previous bounds on both stick number and equilateral stick number by 3. Each knot is shown in orthographic perspective from the direction (3,1,1).

Organization

In building up to Theorems1 and2, we start with background on stick number and equilateral stick number in Section 2 and a review of new approaches to sampling random polygons using symplectic geometry in Section 3. The latter section may be of independent interest, since it gives a concise and reasonably self-contained exposition of the symplectic approach to sampling conﬁned polygons introduced in [16]. We describe our computational pipeline in Section 4 and give a more detailed statement of our results in Section 5, which is supplemented by the tables in the appendices. Finally, we pose some questions and mention some directions for future inquiry in Section 6. This is work that grew out of the ﬁrst author’s master’s thesis [22], which was based on smaller ensembles of random knots.

2 Stick Number

A tame, piecewise-linear embedding S1 ,→ R3 is called a polygonal knot or a stick knot. Any stick knot consists of a ﬁnite number of straight segments. Deﬁne the stick number of a knot K, denoted stick(K), to be the minimum number of linear segments necessary to construct a polygonal version of K. The equilateral stick number, denoted eqstick(K), is the minimum number of congruent segments necessary to construct a polygonal version of K, and of course eqstick(K) ≥ stick(K) for any knot type K. By construction the stick number and equilateral stick number are knot invariants, though not ones that are generally easy to compute. The stick number gives some measure of the inherent geometric (as opposed to purely topological) complexity of a knot, and displays similar behavior as other geometric knot invariants like rope length. For example, the 819 knot has shorter rope length than any 7-crossing knot [6], and it also has smaller stick number than any 7-crossing knot (see Appendix A).

The stick number invariant was deﬁned by Randell [45], who showed that stick(31) = 6 and stick(41) = 7. Randell also showed that every other nontrivial knot must have stick number at least 8, which was an

4 early example of a (in this case lower) bound on stick number. The best general bounds on stick number come from combining work of Negami [42], Huh and Oh [26], and Calvo [11]: Theorem 3 (Negami [42], Huh and Oh [26], and Calvo [11]). For a nontrivial knot K with crossing number c(K), 7 + p8c(K) + 1 3 ≤ stick(K) ≤ (c(K) + 1). 2 2 3 The upper bound can be improved to 2 c(K) for non-alternating prime knots. The lower bound also applies to the equilateral stick number eqstick(K) ≥ stick(K), but the best known theoretical upper bound, due to Kim, No, and Oh [32], is somewhat worse: eqstick(K) ≤ 2c(K) + 2 in general, and eqstick(K) ≤ 2c(K) − 2 for non-alternating prime knots. These bounds are typically quite loose: both inequalities in Theorem 3 are tight for the trefoil and ﬁgure eight knots, and the lower bound is tight for the 819 and 820 knots, but otherwise none of these bounds on stick(K) and eqstick(K) are saturated for any prime knot with 10 or fewer crossings, and it would be somewhat surprising if they were ever saturated by any more complicated knots. Tighter bounds can be proved for more restricted classes of knots. For example: Theorem 4 (Huh, No, and Oh [25]). For any 2-bridge knot K with c(K) ≥ 6,

stick(K) ≤ c(K) + 2.

This bound is sharp for all 6- and 7-crossing knots, and is the best upper bound known for the knots 81−4, 86−9, 811−14, 96, 918, 923, and 1037, as well as for essentially all 2-bridge knots with more than 10 crossings except 11a191, which has 11-stick realizations, and 11a77,89,90,110,111,117,140,154,166,174,177,178,180,190,203,224,247 and 12a537, for which we have observed 12-stick examples. Even better than an upper bound, the exact stick numbers are known for an inﬁnite collection of torus knots: Theorem 5 (Adams et al. [1], Jin [28], Bennett [7], Adams and Shayler [2], and Johnson et al. [29]). For relatively prime 2 ≤ p < q < 3p, the stick number of the (q, p)-torus knot is 2q if q < 2p and 4p if 2p < q < 3p.

In particular, it follows from Theorem 5 that stick(819) = 8, stick(10124) = stick(15n41,185) = 10, and stick(14n21,881) = stick(16n783,154) = 12. In a different direction, Calvo [10, 12] did a deep analysis of the space of 8-stick knots, showing:

Theorem 6 (Calvo [10, 12]). The only knots with stick(K) ≤ 8 are 31, 41, 51, 52, 61, 62, 63, 819, 820, 31#31, and 31# −31.

Said another way, stick(K) ≥ 9 for any knot K not in the above list, and if we encounter such a knot which can be realized as an equilateral 9-gon, then we can immediately conclude that stick(K) = eqstick(K) = 9. In practice, most useful upper bounds on the stick numbers of knots come from examples: after all, if you can ﬁnd a representative of a knot type K with n sticks, then certainly stick(K) ≤ n. Meissen [35] found 9-stick representatives of all 7-crossing knots which, combined with Calvo’s theorem, implies these knots all have stick number 9, and Millett [36, 37] and Calvo and Millett [13] sampled millions of random polygons to ﬁnd representatives of most knots up to 9 crossings with relatively low stick numbers.

5 Rob Scharein obtained much better results in his thesis [49] using a combination of stochastic displace- ments of vertex positions and opportunistic deletions. Rawdon and Scharein [47] extended this approach to the equilateral stick number, and the Scharein and Rawdon–Scharein bounds have remained the gold standard for almost two decades, especially since they are easily referenced online at http://www.colab. sfu.ca/KnotPlot/sticknumbers/ and examples achieving the bounds are built into Scharein’s software KnotPlot [50]. Despite his protestations that he had “not been greatly concerned with efficiency” [49, p. 146], Scharein’s approach gave better results than Calvo and Millett’s Monte Carlo explorations because it was more efficient: the subset of all 9-stick knots which form, say, a 935 knot is so tiny that one needs a truly huge number of random samples to encounter a point in it, whereas Scharein was starting from a conformation known to be a 935 and optimizing it through knot-type-preserving modifications. However, recent advances in (equilateral) polygon sampling algorithms make it possible to uniformly sample random equilateral stick knots in very tight confinement, and to do so quickly enough that generating hundreds of billions of samples is now feasible. Hence, this is a good time to revisit the Monte Carlo approach to finding upper bounds on stick numbers.

3 Symplectic Geometry and Sampling Conﬁned Polygons

Stick knots – which is to say, piecewise-linear embeddings of the circle in 3-space – are of interest in polymer physics and molecular biology, where they appear as models of ring polymers in solution or in melt. The survey [44] gives a nice overview of applications of these models in physics and biology. In this context a piecewise-linear embedding of a circle is usually called a polygon and there has been a lot of work done on developing efﬁcient algorithms for sampling random polygons. Since a polygon is just a stick knot, we can use such an algorithm to generate large ensembles of random stick knots and look for examples with fewer sticks than previously observed. The algorithm we will use is based on understanding the geometry of the space of all possible conformations of equilateral polygons up to rigid motions. To get nice coordinates on this space, we will interpret an 3 n-gon as an ordered collection of edge vectors ~e1,...,~en ∈ R rather than as a list of vertices. In particular, this gives a translation-invariant representation of a polygon.

A list of vectors forms a polygon (as opposed to an open chain) if and only if ~e1 + ··· +~en =~0, and the 2 3 polygon is equilateral if k~eik = 1 for all i = 1,...,n. Hence, using S to denote the unit sphere in R , we will identify the collection of equilateral n-gons up to translation as the subset of S2 × ··· × S2 consisting of | {z } n those n-tuples of unit vectors which sum to zero. This condition is clearly preserved by the diagonal action of the rotation group SO(3), and passing to the quotient produces the space Pol(n) of equilateral n-gons in R3 up to rigid motions. When n is even this SO(3) action is not quite free – degenerate polygons which lie on a line are ﬁxed by any rotation around the line – so Pol(n) is not always a manifold, but it has at worst ﬁnitely many isolated singular points. Moreover, this construction turns out to be nothing more than a symplectic reduction of S2 × ··· × S2 by the diagonal SO(3) action [30], and hence Pol(n) has a natural symplectic structure induced by the standard symplectic structure on S2 × ··· × S2. In fact, Pol(n) is more than merely symplectic or even Kahler:¨ an open, dense subset Pol(n)◦ of Pol(n) is toric [30]. A simple dimension count shows that dimPol(n) = 2n − 6 = 2(n − 3) since ~e1 + ··· +~en =~0 is a codimension-3 condition and SO(3) is 3-dimensional, so being toric means that there is a Hamiltonian

6 v2 v6

v1 d1 v3 d3 d2

v5 v4

v5 v5

v2 v2 θ3

v6 v6 θ2 v1 θ1 v1

v4 v4

v3 v3

Figure 3: The fan triangulation and action-angle coordinates. The n − 3 chords determining the fan triangulation are shown top left. The di determine the dashed lengths in the n − 2 triangles coming from the triangulation; this, together with the solid sides, which are all unit length, uniquely determine the triangles, and the triangulation tells how to glue them together. The θi determine the dihedral angles of the gluings and removing the interiors of the triangles along with the dashed edges produces an n-gon. Starting instead from the bottom right shows how to extract the action-angle coordinates from a particular n-gon.

◦ ◦ n− (n − 3)-torus action on Pol(n) with corresponding moment map µ : Pol(n) → R 3. Consequently, so- called action-angle coordinates give a global coordinate system on Pol(n)◦ which is well-adapted to random sampling. Speciﬁcally, a choice of triangulation of an abstract n-gon – which is to say, a choice of n − 3 non- intersecting segments connecting non-adjacent vertices of an abstract, convex, planar n-gon – induces an action of the (n − 3)-torus. In what follows, we focus on the fan triangulation shown in Figure 3, which connects all non-adjacent vertices to a chosen root vertex. Any triangulation of an n-gon consists of exactly n − 3 chordal segments and produces n − 2 triangles. It is helpful to visualize an n-gon in space as the boundary of a piecewise-linear surface whose faces are exactly these triangles. Each factor of the (n−3)-torus acts by bending the surface around one of the chords and the fact that the chords do not intersect ensures that these actions commute.

The conserved quantities of this action are exactly the lengths d1,...,dn−3 of the chords, and the conju-

7 gate variables θ1,...,θn−3 are the dihedral angels between the two triangles meeting at a given chord. These are the action-angle coordinates induced by the Hamiltonian torus action. When one of the di = 0, neither the circle action nor the corresponding θi make sense, emphasizing that the torus action and the action-angle ◦ coordinates only make sense on the open, dense subset Pol(n) for which all di > 0.

There are no restrictions on the θi, but, since the edges of the original n-gon are all length 1 and di is an edge of two different triangles, the di satisfy a collection of triangle inequalities which can be simpliﬁed as

1 ≤ di + di+1 0 ≤ d1 ≤ 2 0 ≤ dn−3 ≤ 2, (1) −1 ≤ di − di+1 ≤ 1 where the inequalities in the middle column apply for i = 1,...,n − 4. n−3 Let Pn ⊂ R be the convex polytope determined by the inequalities in (1). Given (d1,...,dn−3) ∈ Pn n−3 and any (n − 3)-tuple of angles (θ1,...,θn−3) – which we can think of as a point in the (n − 3)-torus T – we can build a unique element of Pol(n) as illustrated in Figure 3, ﬁrst by building the n − 2 triangles of the triangulation, and then gluing them together according to the θi. The main result of [16] is that, from a n−3 measure-theoretic perspective, Pn × T and Pol(n) are equivalent: n−3 Theorem 7 (Cantarella and Shonkwiler [16]). With respect to the uniform measure on Pn × T and the n−3 natural measure on Pol(n), the reconstruction map Pn × T → Pol(n) deﬁning action-angle coordinates is measure-preserving.

n−3 In particular this implies that any algorithm for randomly sampling points from Pn × T gives an algorithm for sampling random equilateral n-gons. Since it is easy to generate independent random angles, n−3 and hence to sample T , the only challenge is to sample the convex polytope Pn. While the paper [15] gives an algorithm for directly sampling Pn, this approach has not been generalized to the case of confined polygons which we are interested in, so instead we will use the hit-and-run Markov chain introduced by Boneh and Golan [9] and Smith [51] for sampling arbitrary convex polytopes (see also Andersen and Dia- conis’ excellent survey [3]). The idea of hit-and-run is simple: given ~p ∈ Pn, choose a random direction~v; the intersection of the line through ~p containing the direction ~v with Pn is a connected line segment from which we choose the next point in the Markov chain uniformly at random. See Figure 4. n−3 To get an ergodic Markov chain on Pn × T (and hence on Pol(n)), we will mix hit-and-run steps on n−3 Pn and random samples from T as follows. At each step, flip a coin: if heads, iterate hit-and-run 10 2 n−3 times on Pn; if tails, randomly sample a new point (θ1,...,θn−3) from T . This algorithm was dubbed Toric Symplectic Markov Chain Monte Carlo (TSMCMC) and proved to be ergodic in [16]. Unfortunately, knots are quite rare among n-gons when n is small.3 For example, we generated 1 billion random equilateral 10-gons using the TSMCMC procedure described above and only found 6,315,897 (0.63%) nontrivial knots, including only three samples which were 10-crossing prime knots; given the er- godicity of TSMCMC, this gives a decent estimate for the true probability of these knots. For comparison, we know from Appendix A that among the 10-crossing knots there are at least 71 with equilateral stick number ≤ 10. Since more complicated knots tend to be more condensed [43, 40, 20, 19], we expect that sampling polygons in rooted spherical confinement – meaning that the entire polygon is required to be contained in a

2At least in high dimensions, a single hit-and-run step is unlikely to move very far, so it is preferable to do several steps at once, though not too many since hit-and-run steps are much more expensive than sampling from the torus. Our experience is that 10 steps provides a good balance. 3The best theoretical result along these lines is due to Hake [23], who proved that the probability that a random equilateral hexagon 14−3π 1 1 is knotted is bounded above by 192 < 42 ; empirically, the fraction of nontrivial knots among random hexagons is close to 10,000 .

8 t0 [ ~p ~p +t~v

t1 ]

Figure 4: A single hit-and-run step starting from a point ~p ∈ P6 (•). Choosing a random direction produces a unique line through ~p which intersects P6 at the points ~p+t0~v and ~p+t1~v. Choosing t uniformly on [t0,t1] produces the next step ~p +t~v (◦) in the Markov chain. small ball centered at a given root vertex – will boost the probability of encountering complicated knots. To that end, let Pol(n;R) be the subset of n-gons contained in a sphere of radius R centered at the root vertex. n When R ≥ 2 we have Pol(n;R) = Pol(n), but for R very close to its minimum value of 1 (since the second vertex is always at distance 1 from the root vertex) we expect Pol(n;R) to have a much higher fraction of complicated knots than Pol(n). Indeed, as we will see, sampling 10-gons in very tight conﬁnement produces about 7.1% nontrivial knots, and approximately one prime knot with 10 or more crossings for every 1.9 million samples. Fortunately, the TSMCMC algorithm described above can easily be adapted to give an ergodic Markov chain on Pol(n;R) for any R. The constraint that an n-gon lies in a sphere of radius R centered at the root n−3 vertex simply means that each di ≤ R. Adding these inequalities to (1) produces the polytope Pn(R) ⊂ R , n−3 and the analog of Theorem 7 says that Pn(R) × T and Pol(n;R) are measure-equivalent. Therefore, the TSMCMC algorithm described above, when applied to Pn(R) rather than Pn, yields an ergodic Markov chain on Pol(n;R). Here is pseudo-code for this algorithm, taking as input the current position (~p,~θ) ∈ n−3 4 Pn(R) × T with parameters 0 < β < 1 and γ ∈ N:

4 1 As described above, we use β = 2 , so that hit-and-run steps and torus samples are equally likely, and γ = 10, so that we iterate hit-and-run 10 times whenever we choose to do a hit-and-run step.

9 function TSMCMC(~p, ~θ,β,γ) prob = UNIFORM-RANDOM-VARIATE(0,1) if prob < β then for i = 1 to γ do ~v = RANDOM-DIRECTION-IN-DIMENSION(n − 3) (t0,t1) = FIND-INTERSECTION-ENDPOINTS(Pn(R),~p,~v) t = UNIFORM-RANDOM-VARIATE(t0,t1) ~p = ~p +t~v end for else for i = 1 to n − 3 do θi = UNIFORM-RANDOM-VARIATE(−π,π) end for end if return (~p, ~θ) end function

This algorithm depends on the functions UNIFORM-RANDOM-VARIATE(a,b), which produces a (pseudo- )random number uniformly between a and b,RANDOM-DIRECTION-IN-DIMENSION(d), which produces d a vector uniformly at random on the unit sphere in R , and FIND-INTERSECTION-ENDPOINTS(P,~p,~v), which produces the endpoints of the intersection of the convex polytope P with the line through ~p ∈ P in the direction~v. For our experiments we used the version of this algorithm implemented in Ashton, Cantarella, and Chapman’s free and open-source C library plcurve [5].

4 Identifying Knot Types

TSMCMC gives us an efficient algorithm for generating random equilateral polygons in confinement, so our task is in principle simple: generate vast quantities of, say, 10-gons, compute their knot types, and check to see if any are knots not previously known to have (equilateral) stick number less than 11. Therefore, we need a fast, reliable strategy for determining the knot types of hundreds of billions of polygons. Our identification pipeline begins by using the classifier contained in plcurve. While fast, this method of classification has significant shortcomings. Identification is based solely on the HOMFLY polynomial of the given knot, even though this invariant does not uniquely identify knots (e.g. 51 and 10132 have the same HOMFLY polynomial). Furthermore, the plcurve classifier will only identify up to 10-crossing knots, making it unsuitable to detect many knots which we expect to generate. This method of identification main- tains an important place in our classification pipeline, however, because it is sufficient to uniquely identify roughly one half of knots with crossing number 10 or less (including the trivial knot, which represents the vast majority of stick knots generated) and it performs this identification very efficiently. Knots which cannot be definitively classified by plcurve require more robust identification. Toward this end, we integrated the package pyknotid [52] as a follow-up step in our classification pipeline. This package contains a database consisting of many invariants calculated on prime knots with crossing number 15 and below. Identification can be done by computing invariants for a given knot and then checking against

10 the database. In particular, we used pyknotid to identify knots based on their HOMFLY polynomial, hyperbolic volume, and minimum number of crossings. HOMFLY polynomials were calculated by plcurve and hyperbolic volume by SnapPy [18] (via pyknotid). A maximum crossing number was obtained by observing the number of crossings in a default projection of a given knot. Since the pyknotid database only contains the hyperbolic volumes of knots up to 11 crossings, we augmented the database with hyperbolic volumes of knots up to 15 crossings taken from SnapPy. While even this more robust check does not guarantee definitive identification (e.g. mutant knot pairs have the same HOMFLY and hyperbolic volume) it is sufficient for every knot up to 10 crossings5 and many higher-crossing knots. It is important to note that the database queries used in pyknotid identification can be prohibitively slow when identifying hundreds of billions of knots. One critical improvement that we made was to manually add a database index to the columns of the pyknotid invariant database which we were using. This data structure improves query times significantly at a cost of additional storage space and more expensive database writes. However, since the invariant database is static and relatively small, these tradeoffs were well worth the additional speed. We verified this pipeline by plugging in all 249 of Rawdon and Scharein’s minimal equilateral stick knot examples, obtained from KnotPlot. Up to some historical labeling confusion,6 our identification agreed with theirs, verifying that our pipeline is capable of correctly identifying all knots up to 10 crossings. In a very few cases, the above procedure was not able to uniquely identify knot type. When it failed, pyknotid typically returned a list of possibilities; for example, 39 of our 11-gons were identified as either the Kinoshita–Terasaka knot (11n34) or the Conway knot (11n42), but, since these knots are mutants and hence have the same HOMFLY polynomial and hyperbolic volume, they could not be distinguished. In such cases, we computed Dowker–Thistlethwaite codes using KnotPlot [50] and then plugged these into the KnotFinder function on KnotInfo [17], which in turn is based on Knotscape [53]. There were only 889 polygons that fell into this group and KnotFinder was able to identify most, while we identified a few others in ad hoc fashion: in the end, there were only 59 polygons, all of them 12-gons, for which we could not determine the knot type out of the 220 billion polygons we generated. It seems likely that many of these are knots with ≥ 16 crossings, which we would not expect to be able to identify systematically. We also used this process for verification: we loaded the coordinates of all knots mentioned in The- orems1,2, and 11 into KnotPlot, computed DT codes, and then plugged those DT codes into Knot- Info’s KnotFinder. This was not redundant since none of these knots were originally identified by the step in the previous paragraph. Nonetheless, this verification is not entirely independent of our primary plCurve/pyknotid/SnapPy pipeline since some of KnotInfo’s data seems to come from the Knot At- las [33], which serves as the basis for pyknotid’s database, but it was nonetheless reassuring that in each case this second approach verified the knot type identification. The code used to generate and identify each stick knot, along with a summary of the data generated for this paper, is available as part of our stick-knot-gen project [21].

5Under the reasonable assumption that, for knots constructed from so few segments, we are unlikely to encounter a knot with ≥ 16 crossings and the same HOMFLY polynomial and hyperbolic volume as one of the knots up to 10 crossings we are interested in. 6 “Unfortunately, the diagrams of 1083 and 1086 given by Rolfsen do not correspond to their Conway notation or Alexander polynomial—they are interchanged” [24]. This was only corrected in the 2003 edition of Rolfsen’s book – after Rawdon and Scharein’s paper – so it’s not surprising that our pipeline identiﬁed Rawdon and Scharein’s 1083 as a 1086 and identiﬁed their 1086 as a 1083. This is an issue that researchers in the broader computational knot theory community should be aware of: for example, the invariants computed by SnapPy for 1083 match the invariants in the Knot Atlas [33], pyknotid, and KnotInfo [17] for 1086, and conversely the SnapPy invariants for 1086 agree with the Knot Atlas, pyknotid, and KnotInfo invariants for 1083, so whether you identify a given knot as 1083 or 1086 may depend on which software you use. We have used the Knot Atlas naming convention in this paper.

11 � � Crossings R = 1.01 R = 1.1 R = 1.5 0 46,435,788,199 9,457,101,359 9,776,082,950 � 3 3,045,498,555 473,763,655 204,448,942 4 355,759,490 49,671,263 15,247,185 ��-� � 5 141,328,728 17,008,560 3,784,326 � 6 15,054,589 1,753,019 325,635 7 676,177 79,926 13,097 � 8 3,046,297 322,455 50,839 9 238,874 25,119 3,374 � 10 22,211 2,608 292 �� -� � 11 4,364 500 42 � 12 345 28 2 13 85 7 1 �� 14 0 0 0 �� 15 1 1 0 ��-� Composite 2,582,085 271,500 43,315 �� Total 50,000,000,000 10,000,000,000 10,000,000,000 �� Figure 5: Left: A log plot of the probability of different crossing numbers among random 10-gons sampled from rooted spherical conﬁnement of radii R = 1.01,1.1,1.5. Only prime knots are included in this plot. Note the inversion between 7- and 8-crossing knots, which is mostly due to the knots 819 and 820. These knots both have stick number 8, whereas all 7-crossing knots have stick number 9, and each of these knots individually appeared more than 1.5 times as often as all 7-crossing knots put together at all three conﬁne- ment radii. Right: The complete data for 10-gons. Keep in mind that we generated 5 times as many 10-gons at R = 1.01 as at the other two radii. Complete frequency data can be downloaded in CSV format from the stick-knot-gen project [21].

5 Results

We now have all the necessary tools in place to generate large ensembles of equilateral polygons in very tight conﬁnement and to identify their knot types. Given the imprecision of ﬂoating point arithmetic, the “equilateral polygons” we generate on the computer will only be approximately equilateral, so we need some guarantee that, if we generate an approximately equilateral polygon of a given knot type, there exists a true equilateral polygon of the same knot type. The following theorem does just that:

Theorem 8 (Millett and Rawdon [39]). Let K be an n-stick knot where Li is the length of the ith edge. Let µ(K) denote the minimum distance between any two non-adjacent edges. If

µ(K) µ(K)2 |L − 1| < min , i n 4 for all 1 ≤ i ≤ n, then there exists an unit-edge equilateral stick knot equivalent to K.

We focused on generating 9-, 10-, and 11-gons, since most stick numbers of knots up to 10 crossings lie in this range. Since we are free to choose the conﬁnement radius R, we started by generating 10 billion n-gons for n = 9,10,11 in rooted spherical conﬁnement of radii R = 1.01,1.1,1.5 to get a sense for which regime produced the highest concentration of complicated knots. Not surprisingly, R = 1.01 was the clear

12 winner; see Figure 5. Consequently, we focused the bulk of our effort on generating polygons in this very tight confinement regime, producing a total of 50 billion each of equilateral 9-, 10-, and 11-gons in confinement radius R = 1.01. We also generated 10 billion equilateral 12-gons in confinement radius 1.01 in the (justified!) hope that we would find 12-stick representatives of any remaining knots. Among all these random equilateral polygons, a tiny fraction realized a knot type with fewer equal- length sticks than previously observed. For each such knot type, we took the sample maximizing the minimal distance µ between non-adjacent edges, and then stochastically applied crankshaft rotations [36] (which preserve edge lengths) in search of polygons of the same knot type maximizing µ. The coordinates of the vertices of each of the resulting optimal polygons are recorded in the data directory of stick-knot-gen [21]. For every conformation, µ(K) µ(K)2 |L − 1| < 10−2.69 min , , i n 4 so the hypothesis of Theorem 8 is easily satisfied, and we conclude that there is a true equilateral polygon with the same number of sticks representing each knot type we found.

In particular, since we observed equilateral 9-gon versions of the knots 935, 939, 943, 945, and 948, Theorem 6 and the following discussion implies Theorem 1, which we restate:

Theorem 1. The stick number and equilateral stick number of each of the knots 935, 939, 943, 945, and 948 is exactly 9.

We recall a result of Randell [46] relating stick number and the superbridge index sb(K) of a knot K, which is the minimum over all embeddings of K of the maximum number of local maxima in any projection to a line:

1 Theorem 9 (Randell [46]). For any knot K, sb(K) ≤ 2 stick(K).

Combined with Theorem 1, this implies that each of the knots 935, 939, 943, 945, and 948 has superbridge index ≤ 4. Since the superbridge index of a knot is strictly greater than the bridge index [34] and each of these knots has bridge index equal to 3 [17], we get the following corollary:

Corollary 10. Each of the knots 935, 939, 943, 945, and 948 has superbridge index equal to 4.

We also discovered a number of knots realizable with 10, 11, or 12 equilateral sticks that had not previously been observed:

The equilateral stick number of each of the knots 1076 and 1080 is less than or equal to 12. In particular, all knots up to 10 crossings have equilateral stick number ≤ 12.

13 11n74 11n81

Figure 6: These 11-stick examples of 11n74 and 11n81 prove that these knots have superbridge index 5. Each knot is shown in orthographic perspective from the direction of the positive z-axis.

In addition to the knots listed above, among the 10-stick knots we generated were 195 different prime knots with 11 or more crossings, including the 15-crossing knots 15n41,127 and 15n59,007; among the 11- stick knots we generated were 1212 different prime knots with 11 or more crossings, including eight 16- crossing knots; and among the 12-stick knots were at least 1888 different prime knots with 11 or more crossings. Coordinates for all of these knots with more than 10 crossings can be found in the supplemental data provided in stick-knot-gen [21]. We observed a number of 4-bridge knots among the 11-gons (see Figure 6 for two examples), meaning these knots have superbridge index 5:

Theorem 11. Each of the knots 11n71, 11n73, 11n74, 11n75, 11n76, 11n78, and 11n81 has superbridge index equal to 5 and stick number either 10 or 11.

Proof. By Theorem 9 these knots satisfy 2sb(K) ≤ stick(K) ≤ 11. Since superbridge index is strictly greater than bridge index and each of these knots has bridge index 4 [41], the result follows.

Coordinates of each of the knots mentioned in Theorems1,2, and 11 are given in Appendix B and can be downloaded from the stick-knot-gen project [21].

6 Conclusion and Open Questions

To the best of our knowledge the information on stick numbers of knots up to 10 crossings contained in Appendix A represents the state of the art, but it likely remains incomplete. For example, we know from Rawdon and Scharein’s work [47] that there are equilateral 10-stick realizations of the knots 916, 10109, 10113, and 10114 and equilateral 11-stick realizations of the knots 101 and 10123, but we never saw any examples with these stick numbers among the 140 billion equilateral 10- and 11-gons we generated. Of course, Rawdon and Scharein’s bounds on these 6 knots are still given in Appendix A, but it seems implausible that these are the only knots achievable with 10 or 11 sticks that we failed to generate.

14 We reiterate the viewpoint that generating random polygons in confinement is a strategy for boosting the probability of complicated knots. We generated equilateral stick knots in rooted spherical confinement because this is the only confinement regime in which we know of a strategy for ergodically sampling equilateral polygons, but there is no particular reason to believe that rooted spherical confinement is optimal for producing complicated knots. Aesthetically, unrooted spherical confinement seems preferable, with the substantial side benefit that an understanding of the probability distribution of random equilateral polygons in unrooted spherical confinement would have important applications to the packing of DNA in viral capsids and other biological problems [4, 27]. Since we wanted to see as many complicated knots as possible, we have mostly focused on knots in very tight confinement. There is no particular reason to believe, though, that any knot which is constructible with n equilateral sticks is necessarily constructible with n equilateral sticks in any given confinement radius. Indeed, Rawdon and Scharein’s 11-stick equilateral 10123 cannot be put in rooted spherical confinement of radius less than 1.1638, the largest of all of their examples, which may explain why we did not observe 10123 among our 11-gons. Similarly, we observed 41#62 and 41#63 knots among the 10 billion 11-stick knots in rooted spherical confinement of radius 1.1 that we generated, but not among our 50 billion 11-stick knots in confinement radius 1.01. Relatedly, composite knots remain relatively poorly understood in the context of stick number, and we know of no definitive table of stick numbers for low-crossing composite knots. The TSMCMC approach to generating equilateral polygons described in Section 3 generalizes to arbitrary spaces of polygons with given edge lengths in rooted spherical confinement. If we want to generate n-gons with edgelengths given by (r1,...,rn) with ri ≥ 0 in rooted spherical confinement of radius R, we must slightly modify the inequalities in (1), but otherwise the algorithm is exactly the same and is still ergodic. Since equilateral polygons are more flexible than any other polygons of fixed edge lengths [14, 31], we do not expect that sampling random polygons with other fixed edge lengths will produce generally better bounds on stick number than sampling random equilateral polygons. However, it remains an open question whether stick(K) = eqstick(K) for all knots K, so sampling other fixed edge length spaces might help to shed light on this problem. We know of only two examples of knots up to 10 crossings for which the upper bounds on stick number and equilateral stick number differ: 929 and 1079. Rawdon and Scharein’s equilateral 12-stick 1079 cannot be put in rooted spherical confinement of radius less than 1.1367, second only to their 10123 mentioned above, suggesting that an equilateral 11-stick example, if it exists, may not occur in the tight confinement regime we have focused on. 929 is particularly interesting since its stick number is exactly 9, and hence proving that the equilateral stick number of 929 is 10 would immediately imply that stick number and equilateral stick number are distinct invariants. More generally, any theoretical results on stick number or equilateral stick number, even for individual knots, would be interesting and welcome additions to the field. In a companion paper [8], we use the relationship between bridge index, superbridge index, and stick number to determine the exact stick number of the knots 13n592 and 15n41,127.

Acknowledgments

We are grateful to Colin Adams, Ryan Blair, Jason Cantarella, Harrison Chapman, Tetsuo Deguchi, Kate Hake, Gyo Taek Jin, Ken Millett, Eric Rawdon, and Erica Uehara for helpful conversations about random polygons and stick knots. This work was partially supported by a grant from the Simons Foundation (#354225, CS).

15 References

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19 A Stick Number Bounds

We give the best known upper bounds on stick number for all prime knots up to 10 crossings, including references for where the given bound was proved. Bounds marked with an asterisk (∗) are known to be the exact stick number of the knot. Two knots, 929 and 1079, have bounds marked with a dagger (†): these are the two knots for which the best bounds on stick number and equilateral stick number differ. In both cases, the equilateral stick number bound is 1 greater than the bound listed in this table. Coordinates of equilateral versions of each of these knots realizing the best known bound can be downloaded either as tab-separated text ﬁles or in an SQLite database from the stick-knot-gen project [21].

K stick(K) bound K stick(K) bound K stick(K) bound ∗ ∗ 01 3 817 9 [10, 47] 928 10 [47] ∗ ∗ ∗† 31 6 [45] 818 9 [11, 47] 929 9 [10, 47] ∗ ∗ 41 7 [45] 819 8 [1, 28, 38] 930 10 [47] ∗ ∗ 51 8 [42, 35] 820 8 [42, 35] 931 10 [47] ∗ ∗ 52 8 [28] 821 9 [10, 35] 932 10 [47] ∗ 61 8 [42, 35] 933 10 [47] ∗ 91 10 [47] ∗ 62 8 [42, 35] 934 9 [10, 47] ∗ 92 10 Thm. 2 ∗ 63 8 [42, 35] 935 9 [10], Thm. 1 93 10 Thm. 2 ∗ 936 11 [47] 71 9 [10, 35] 94 10 [47] ∗ 937 10 [47] 72 9 [10, 35] 95 10 [47] ∗ 938 10 [47] 73 9 [10, 35] 96 11 [47] ∗ ∗ 939 9 [10], Thm. 1 74 9 [10, 35] 97 10 [47] ∗ ∗ 940 9 [10] 75 9 [10, 35] 98 10 [47] ∗ ∗ 941 9 [10] 76 9 [10, 35] 99 10 [47] ∗ ∗ 942 9 [10] 77 9 [10, 35] 910 10 [47] ∗ 943 9 [10], Thm. 1 911 10 Thm. 2 ∗ 81 10 [47] 944 9 [10, 47] 912 10 [47] ∗ 82 10 [47] 945 9 [10], Thm. 1 913 10 [47] ∗ 83 10 [47] 946 9 [10] 914 10 [47] ∗ 84 10 [47] 947 9 [10, 47] 915 10 Thm. 2 ∗ 85 10 [47] 948 9 [10], Thm. 1 916 10 [47] ∗ 86 10 [47] 949 9 [10, 47] 917 10 [47] 87 10 [47] 918 11 [47] 101 11 [47] 88 10 [47] 919 10 [47] 102 11 [47] 89 10 [47] 920 10 [47] 103 11 Thm. 2 810 10 [47] 921 10 Thm. 2 104 11 [47] 811 10 [47] 922 10 [47] 105 11 [47] 812 10 [47] 923 11 [47] 106 11 Thm. 2 813 10 [47] 924 10 [47] 107 11 Thm. 2 814 10 [47] 925 10 Thm. 2 108 10 Thm. 2 815 10 [47] ∗ 926 10 [47] 109 11 [47] 816 9 [10, 47] 927 10 Thm. 2 1010 11 Thm. 2

20 K stick(K) bound K stick(K) bound K stick(K) bound

1011 11 [47] 1043 11 Thm. 2 1075 11 Thm. 2 1012 11 [47] 1044 11 Thm. 2 1076 12 Thm. 2 1013 11 [47] 1045 11 [47] 1077 11 Thm. 2 1014 11 [47] 1046 11 Thm. 2 1078 11 Thm. 2 † 1015 11 Thm. 2 1047 11 Thm. 2 1079 11 [47] 1016 10 Thm. 2 1048 10 [47] 1080 12 Thm. 2 1017 10 Thm. 2 1049 11 [47] 1081 11 [47] 1018 11 Thm. 2 1050 11 Thm. 2 1082 11 Thm. 2 1019 11 [47] 1051 11 Thm. 2 1083 10 Thm. 2 1020 11 Thm. 2 1052 11 [47] 1084 11 Thm. 2 1021 11 Thm. 2 1053 11 Thm. 2 1085 10 Thm. 2 1022 11 Thm. 2 1054 11 Thm. 2 1086 11 [47] 1023 11 Thm. 2 1055 11 Thm. 2 1087 11 [47] 1024 11 Thm. 2 1056 10 Thm. 2 1088 11 [47] 1025 11 [47] 1057 11 Thm. 2 1089 11 [47] 1026 11 Thm. 2 1058 12 [47] 1090 10 Thm. 2 1027 11 [47] 1059 11 [47] 1091 10 Thm. 2 1028 11 Thm. 2 1060 11 [47] 1092 11 [47] 1029 11 [47] 1061 11 [47] 1093 11 [47] 1030 11 Thm. 2 1062 11 Thm. 2 1094 10 Thm. 2 1031 11 Thm. 2 1063 11 [47] 1095 11 Thm. 2 1032 11 [47] 1064 11 Thm. 2 1096 11 [47] 1033 11 [47] 1065 11 Thm. 2 1097 11 Thm. 2 1034 11 Thm. 2 1066 12 [47] 1098 11 [47] 1035 11 Thm. 2 1067 11 [47] 1099 11 [47] 1036 11 [47] 1068 11 Thm. 2 10100 11 Thm. 2 1037 12 [47] 1069 11 [47] 10101 11 Thm. 2 1038 11 Thm. 2 1070 11 Thm. 2 10102 10 [47] 1039 11 Thm. 2 1071 11 Thm. 2 10103 10 Thm. 2 1040 11 [47] 1072 11 Thm. 2 10104 10 [47] 1041 11 [47] 1073 11 Thm. 2 10105 10 Thm. 2 1042 11 [47] 1074 11 Thm. 2 10106 10 Thm. 2

21 K stick(K) bound K stick(K) bound

10107 10 [47], Thm. 2 10139 10 [47] 10108 10 [47] 10140 10 [47] 10109 10 [47] 10141 10 [47] 10110 10 Thm. 2 10142 10 Thm. 2 10111 10 Thm. 2 10143 10 Thm. 2 10112 10 Thm. 2 10144 10 [47] 10113 10 [47] 10145 10 [47] 10114 10 [47] 10146 10 [47] 10115 10 Thm. 2 10147 10 [47], Thm. 2 10116 10 [47] 10148 10 Thm. 2 10117 10 Thm. 2 10149 10 Thm. 2 10118 10 Thm. 2 10150 10 [47] 10119 10 [47], Thm. 2 10151 10 [47] 10120 10 [47] 10152 11 [47] 10121 10 [47] 10153 10 Thm. 2 10122 10 [47] 10154 11 [47] 10123 11 [47] 10155 10 [47] ∗ 10124 10 [1, 28] 10156 10 [47] 10125 10 [47] 10157 10 [47] 10126 10 Thm. 2 10158 10 [47] 10127 10 [47] 10159 10 [47] 10128 10 [47] 10160 10 [47] 10129 10 [47] 10161 10 [47] 10130 10 [47] 10162 10 [47] 10131 10 Thm. 2 10163 10 [47] 10132 10 [47] 10164 10 Thm. 2 10133 10 Thm. 2 10165 10 [47] 10134 10 [47] 10135 10 [47] 10136 10 [47] 10137 10 Thm. 2 10138 10 Thm. 2

22 B Stick Knot Coordinates

We give coordinates of equilateral realizations of each of the knots mentioned in Theorems1,2, and 11 below. Each row gives the coordinates of a vertex. These coordinates can be downloaded either as tab- separated text ﬁles or in an SQLite database from the stick-knot-gen project [21].

92 921

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.999999999973 -0.000000000238 -0.000000006378 0.999999999988 -0.000000001732 0.000000008667 0.377787294690 0.782848228577 -0.000000004464 0.116912849445 0.469208997027 -0.000000000434 0.481892968809 -0.210260817716 0.053818494146 0.299142422074 -0.499974342149 -0.165758975821 -0.385834243146 0.031557373199 -0.380431949892 0.749383175444 0.375384108431 -0.341914722302 -0.144253062985 0.496729049752 0.471186418115 0.560880577502 -0.293068061976 0.377556240605 0.162700176100 -0.186949875250 -0.190903331479 0.243435681189 0.217582941613 -0.421483364398 0.684883875886 0.311048799061 0.501427156130 0.434924293424 -0.518032959544 0.228286869895 0.276972402756 0.908702492523 -0.188803452286 0.232932098142 0.459738136499 0.172047766022 0.974395069202 0.223759836486 0.022040523880 0.876539844203 0.055871061071 -0.478075648873

93 925

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.999999999959 0.000000006901 0.000000002872 1.000000000000 -0.000000004700 0.000000000423 0.372624758458 0.778717096894 0.000000002526 0.297955295632 0.712132871284 0.000000005690 0.357966555261 0.557523627122 -0.975119777801 0.508816958364 -0.259809494361 -0.104237209739 0.949956765273 0.586793597600 -0.169706306290 -0.248158854990 0.328774125004 0.179588337092 0.233474142980 -0.087214018489 -0.349612908264 0.699925636040 0.351639450261 -0.137607146377 0.769117889647 0.327878773715 0.385767095931 -0.294671078076 0.367559159613 -0.035020894393 0.692196419337 0.577317092949 -0.579563734569 0.425269849180 -0.040398159834 0.526455629381 0.229065705696 0.855448237280 0.261954541320 0.297126127816 0.547143528614 -0.272527443682 0.883542651989 0.355339529733 -0.305100312661 0.596216003363 0.434574530250 0.675034410226

911 927

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000050 0.000000000719 -0.000000005639 1.000000000110 -0.000000004551 0.000000003763 0.070051054207 0.367688670977 0.000000002547 0.073818706276 0.377078516712 0.000000001267 0.108161326136 0.376223269343 -0.999237090356 0.337143126688 -0.567884771228 0.194176812560 0.128417640945 0.525388229801 -0.010632267228 0.329472622438 0.320571578615 -0.264720210545 0.669249891636 -0.315423441298 0.012523948856 1.035060372380 0.015232248216 0.374743486959 0.080331781660 0.286262575316 -0.527058664982 0.286264824178 0.159348536766 -0.272199881776 0.378578125067 0.387716575752 0.422023104114 0.153213377745 -0.050853560207 0.696362140529 0.016208553742 0.340622745128 -0.508820848917 0.359580868476 -0.267310983410 -0.257869824305 0.549670755302 0.812794735357 0.192942423967 0.740850508636 -0.137529788493 0.657439032285

915 935

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000030 0.000000006932 0.000000000155 0.999999999935 -0.000000004489 -0.000000006432 0.111494566011 0.458866102614 0.000000002644 0.319886562581 0.733106885568 -0.000000004192 0.913398426052 -0.121483088140 -0.141933136704 0.558297071061 0.119190194309 -0.752506964721 0.392655562784 0.073301376104 0.689262317371 0.404552532087 -0.127158422011 0.204402109335 0.242928301101 0.098977602409 -0.299131570962 -0.027529078858 0.464069732340 -0.476593302569 0.706185391596 -0.095940489595 0.565391306402 0.722283839107 0.597133655906 0.171538304670 0.723960029669 0.053752620436 -0.423181419823 0.398573339896 -0.273154812344 -0.199688707759 0.218381899324 0.417982538309 0.358948570026 0.579860399875 0.677042860632 -0.453183055204 0.940617915616 0.177316826289 -0.289476562011

23 939 103

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000050 0.000000000947 0.000000006319 1.000000000000 0.000000000000 0.000000000000 0.104465710090 0.444992512795 0.000000003714 0.295099436133 0.709306136348 0.000000000000 0.385853182847 -0.298247876945 -0.606971835169 0.743634443843 -0.171244634576 0.153123109571 0.796515213189 0.452556941574 -0.089623962555 -0.047653955306 0.076707113133 -0.405788869349 0.009741230786 -0.138634877270 0.087799074143 0.676921396905 -0.561236049500 -0.144984191057 0.398256777112 0.381736091648 -0.672641406730 0.550736205384 0.364494842341 0.211526136046 0.839413704804 0.113763413508 0.183846310016 0.651652056677 -0.120179196512 -0.657327745236 0.633086696236 -0.138145067675 -0.761654236076 0.891059518228 0.459376243326 0.121645655791 0.479007477232 -0.405786741957 -0.164197175862 0.181898143352 0.092028546074 -0.979001436236

943 106 0.000000000000 0.000000000000 0.000000000000 0.999999999976 0.000000004022 -0.000000002241 0.123205019372 0.480864394418 -0.000000003121 0.000000000000 0.000000000000 0.000000000000 0.536781299678 -0.425023241338 0.091226373410 1.000000000000 0.000000000000 0.000000000000 0.655686472913 0.511849704238 -0.237605868949 0.076622881037 0.383894121049 0.000000000000 0.130945474415 -0.166080553236 0.277231435907 0.384225465337 -0.256613828862 0.703654898536 0.466858434733 0.773364194053 0.345099321197 0.168821062775 -0.085069515409 -0.257684527411 0.615641998136 0.055456942967 -0.334954081175 0.013953723959 0.357144939045 0.625753350224 0.725959223388 0.255055144373 0.638694042013 0.324207756843 0.193585699194 -0.310724524656 0.767571065652 -0.335789629956 0.412594725254 0.604039392965 0.388524122413 -0.257201497606 -0.185719292546 0.218009978692 0.332040511631 0.729461040842 0.618973278237 0.291133424262

945

0.000000000000 0.000000000000 0.000000000000 0.999999999976 -0.000000003405 0.000000001560 107 0.190799788131 0.587532988957 0.000000003866 0.618532229693 -0.316173690378 0.018952475450 -0.168249315081 0.200679011262 -0.318446525138 0.000000000000 0.000000000000 0.000000000000 0.665648318313 -0.150518614023 0.107317452947 1.000000000000 0.000000000000 0.000000000000 -0.041271805254 0.070599205069 -0.564523933847 0.031034387757 0.247195554753 0.000000000000 0.448339322230 -0.004956385151 0.304137275502 0.433153690970 -0.344914154636 0.698359619416 0.754811851302 0.192729936955 -0.626988229980 0.470489324230 -0.068757795504 -0.262027654448 0.168431571655 0.094843643407 0.677118597898 0.509844174096 0.301821913731 -0.239722408225 0.376735408028 -0.453231490128 0.402287259706 0.891806573865 0.290193977028 -0.024351075313 0.071592959174 -0.152060592797 0.338499474837 0.679248606446 0.641953919712 0.355691573709 948

0.000000000000 0.000000000000 0.000000000000 0.999999999996 -0.000000003564 0.000000005184 0.248818509090 0.660095722600 0.000000004874 0.485153931120 -0.218697628419 0.414569437562 108 0.660586822993 0.407932508387 -0.344744708295 0.723479433911 -0.029769564714 0.552173021844 0.032497435603 -0.280512703542 -0.125817948837 0.000000000000 0.000000000000 0.000000000000 0.550894729598 0.543039852974 0.104454568431 1.000000000000 0.000000001474 0.000000003955 0.814146590780 -0.157417362979 -0.558914217521 0.317011378212 0.730429014105 -0.000000004922 0.366653340016 -0.250196709107 -0.189496354767 0.627561312612 0.438817472656 0.486659313107 0.371113287586 0.688873356349 -0.446992905598 0.395833015069 0.558378615406 0.544147880372 0.643436615623 0.507929649370 -0.423399203151 0.578102914549 0.233720047605 0.536048868314 0.497987557224 0.825317540193 -0.266194197282

24 1010 1018

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.236713421703 0.646060058657 0.000000000000 0.312756934294 0.726427538464 0.000000000000 0.083067046323 0.065888378288 0.799870997493 0.680458110320 -0.203177008275 0.025124327480 0.336900556633 -0.045168792223 -0.160980111640 -0.226772818215 0.210320007757 -0.052026563909 -0.042583906181 0.314217920204 0.691564745487 0.495053855569 0.378212968617 -0.723426747636 0.664727599604 0.406271213498 -0.009318001467 0.484376310065 0.835115589836 0.166025884461 -0.181389636369 -0.126488366052 -0.025219365773 -0.008546411973 0.210754645410 -0.439944907883 0.623612904546 -0.651797537767 0.250500026422 0.171149599288 0.412684529962 0.522829035087 0.441792759643 0.278664764841 -0.067590115413 0.793440265196 0.439084936158 -0.259511904450 0.139532019856 0.485809336725 0.862855783886 -0.079857812413 0.843380434260 -0.531349388731

1015 1020

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.113905380497 0.463504396191 0.000000000000 0.153219825859 0.531942982547 0.000000000000 -0.122662405858 -0.008179859426 -0.849440783965 0.008535087060 -0.404403502899 -0.319877453960 0.461462969379 0.325450558517 -0.109516267390 0.210642827625 0.525004170887 -0.011083287707 0.014576088792 -0.279703585255 -0.768364184769 0.615746552276 -0.208370891073 0.534858092635 -0.141603970788 0.658328832868 -0.459005216028 0.627026972699 0.312587105244 -0.318649680141 0.101154465045 -0.220358757355 -0.047937751225 0.455379230799 0.376610548920 0.664426131463 0.556571132196 0.663683303751 0.057253947484 0.169400011945 0.387883602480 -0.293743375800 -0.191330403038 0.099390079681 -0.292340747648 -0.000931253153 0.258966804641 0.683174082006 0.720579538016 0.413211436303 -0.556795688114 0.858364990240 0.465666964496 0.215322599155

1016 1021

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.999999999986 -0.000000003603 -0.000000006635 1.000000000000 0.000000000000 0.000000000000 0.170126987122 0.557952308449 -0.000000003966 0.152873600923 0.531391441395 0.000000000000 0.701043378871 -0.131832773177 -0.492264491984 0.677578940856 -0.317913440094 -0.058013140976 0.209092261015 -0.077091365888 0.376635661043 0.107136492050 0.415724385633 0.311271238957 0.728773122463 0.437933917337 -0.305039305390 0.615929401065 -0.427586631806 0.484350133111 0.148392593962 -0.375633518675 -0.269451818753 0.416686206671 0.342005321812 -0.122304937265 0.943392667843 0.140481965703 0.049295242929 0.181167997366 0.017801254357 0.793895311442 0.139867814717 -0.088052260750 -0.500359119158 0.587128141122 0.580635225517 0.073885393046 0.848537298960 0.512785648430 -0.130519466055 0.179722861126 -0.278695750243 -0.235260546489 0.948044869340 0.049486939149 0.314264170042

1017 1022

0.000000000000 0.000000000000 0.000000000000 1.000000000130 0.000000000461 0.000000005147 0.000000000000 0.000000000000 0.000000000000 0.222791291659 0.629242897682 0.000000005194 1.000000000000 0.000000000000 0.000000000000 0.446486902003 -0.318564366999 -0.227204006466 0.153386561242 0.532208310076 0.000000000000 0.449121999841 0.603307281376 0.160282406381 0.220071080827 -0.454485174753 0.148287362390 -0.205386109038 0.308428325255 -0.535896835337 0.249610063606 0.455525533908 0.561818447823 0.252426661285 -0.074742809820 0.266342124442 0.194953281050 -0.511142690599 0.311688122093 0.674185090528 0.052223831190 -0.631432512892 0.512673200914 0.361769028037 -0.058553939253 0.480892535438 0.189091456676 0.340115345538 -0.162052422628 0.264886162891 0.673128398497 0.346662501947 0.848704529161 -0.399406724928 0.311802314667 -0.489954816050 0.219612374023 -0.070649665646 0.373085146239 -0.110376271610 0.810798874329 0.467353597502 0.352400057162

25 1023 1030

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.034684780542 0.261087202067 0.000000000000 0.109193226206 0.454382318938 0.000000000000 0.245456758352 -0.096049234001 -0.909960844983 0.732880141801 0.053838927198 0.671252279438 0.224842233420 -0.270024572828 0.074573371203 -0.158540649557 -0.283378805427 0.368509528015 -0.171071128665 0.399210124353 -0.554217158155 0.504887345897 0.398930002775 0.675621861044 0.457491855053 0.043919795462 0.137647971545 0.848110486707 0.228671769972 -0.248071816413 0.024410110808 -0.286911689008 -0.700797454041 0.720801803494 -0.326183235607 0.574076845303 0.759148469928 0.126090756195 -0.162663154916 0.501041766076 0.649351182008 0.580251471035 -0.123194791675 0.305526796885 -0.597718411183 0.524198619185 -0.348042670866 0.648583472348 0.533318181093 -0.438217026749 -0.723558950731 0.765340971759 0.562505020285 0.312795938433

1024 1031

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.059107707270 0.338705910018 0.000000000000 0.156348008746 0.536890414939 0.000000000000 0.197268732252 -0.163592938267 0.853585026923 0.738412822641 -0.239268241351 0.242442349212 0.152985492617 -0.263497378168 -0.140426088496 -0.057834893835 0.352748022659 0.366966712907 -0.244162696994 0.507938799152 0.356725336939 0.778563498045 0.262847525082 -0.173732492378 0.468912601385 -0.010653396558 -0.115064609319 0.678742303281 -0.554060333436 0.394332010798 0.090083300454 -0.298754768476 0.764415896579 -0.056866672375 -0.024747081543 -0.028405397419 0.447169291507 0.589401686127 0.475162189357 0.688686616621 0.634932172636 0.066323557352 -0.382205058471 0.038132057747 0.384387829404 0.572772251293 -0.309173324894 -0.242267870661 0.399296901281 -0.407039576689 0.821511270547 0.721902248429 0.015211402997 0.691827837639

1026 1034

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.348637772012 0.758766925973 0.000000000000 0.168855351506 0.556056268087 0.000000000000 0.681164467759 0.025483919399 -0.593061572592 0.619075796556 -0.333404800147 -0.078489228269 -0.014455223673 -0.089976529398 0.116009742338 -0.038820253294 -0.273370247637 0.672222860416 0.892906488952 0.207717515435 -0.180761176274 0.441374422834 0.005944352041 -0.159279651562 0.195642126755 0.078041720486 -0.885748124674 0.229197152155 0.812816984748 0.392025491164 0.601863980366 0.726675728401 -0.242119559875 0.343861238139 0.121474075591 -0.321344769881 0.163601610177 -0.164584913520 -0.125583887057 -0.257371018547 0.402265465390 0.426770173783 0.839511047571 0.544600959778 0.074920051950 0.572077602092 0.436405785749 -0.130768546170 0.801330839793 -0.235074192887 -0.550099090197 0.851860482027 -0.404071793259 0.333256215325

1028 1035

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.051376909076 0.316408330114 0.000000000000 0.063963886523 0.351903956026 0.000000000000 0.870985927072 -0.090912040437 0.402903429315 0.801592032088 0.018035589819 0.586887240376 0.027249430203 0.381469960764 0.148023126443 0.745075837399 -0.137270072922 -0.399361242948 0.857445338309 0.234123831006 -0.389623478873 0.420297328348 0.602006913136 0.190544222313 0.341180165884 0.670913221877 0.347048293195 0.891684159889 -0.232671268801 -0.094248312084 0.516155685242 -0.294807190072 0.155299359508 0.543478575090 0.285463349641 0.686961895151 0.476422054584 0.630256800555 -0.222427859316 0.196189781642 0.060188373012 -0.223335681936 0.117043028445 -0.046748122969 0.419842371269 -0.101108989012 0.805308228255 0.373665023499 0.625838914318 0.767319867056 0.139806562602 0.585864917390 0.731296751195 -0.349238257159

26 1038 1046

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.999999999950 0.000000003250 -0.000000004981 0.042565739497 0.288651410555 0.000000000000 0.059838434901 0.340728974007 0.000000001238 0.299137109616 -0.339369508064 -0.734684189167 0.264703535774 -0.605763013880 0.249365610990 -0.121891088950 -0.181290973131 0.158482331033 -0.340287133056 0.190224934726 0.229630456974 0.465703686834 0.570153289201 -0.141624150202 0.533012665099 -0.141461219398 -0.127205300341 0.256282770549 -0.258286230760 0.377828293809 0.066351402945 0.564557987927 0.405489870750 -0.042612773453 0.289957461635 -0.403254485111 -0.610515719252 -0.154874930051 0.249710618618 0.428953348474 0.054489027589 0.446557264172 0.219725008840 0.333722310426 -0.018567352869 0.181580887646 0.091383272752 -0.521660521400 0.099849053615 -0.247293907615 0.786448118883 0.987327776594 0.143232232619 0.068325610905 -0.579207847879 0.476112861900 0.661690873211

1039 1047

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.386085532640 0.789372552580 0.000000000000 0.165588160362 0.551141435452 0.000000000000 0.655247224790 0.103454295781 0.676068138924 0.095759985280 -0.446338809330 0.012537433338 0.179942376189 0.296837814552 -0.182237235382 -0.220196478441 0.314826430734 0.578928632380 0.768318812581 -0.481583876430 0.036561393742 0.158765337554 -0.175707043551 -0.205778225385 0.542580893886 0.345796955049 0.550840842839 -0.182817874754 -0.013164147260 0.719911093951 0.654607311786 0.581087379980 -0.414606461589 -0.080856412279 -0.386021037549 -0.202358714475 0.751378847683 -0.137966689974 0.273577027258 0.469085936002 0.383696117709 0.121830912857 0.538737337463 0.128544713730 -0.666505554129 -0.078145737358 -0.443236130962 -0.007477059747 0.738927652261 0.654482042335 0.160122393766 0.335806360494 -0.302821186027 0.891926800553

1043 1050

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.318437803844 0.731760188020 0.000000000000 0.253721842446 0.665634217538 0.000000000000 0.672194137225 -0.077028770377 0.469805148298 0.397863922767 -0.286613247305 0.269161338939 0.680538390430 0.296589141463 -0.457740014694 0.321083854063 0.539467277052 -0.289135883270 -0.008159373750 -0.063659999024 0.171478505689 0.348192874372 0.213848756850 0.655976639300 0.680010984596 0.097180484521 -0.536018417801 0.896641043206 0.371395788823 -0.165231944934 0.563485984663 0.527426984116 0.359140715524 0.189078087270 -0.055272372930 0.398069877480 0.087887770996 0.609917918156 -0.516645551186 0.793570755186 -0.022904445784 -0.397882967687 0.096445407122 0.536177876618 0.480595228735 0.603591791023 0.134421688756 0.571217894229 0.907761080012 0.409683164561 -0.090163885730 0.681996607669 0.683164171035 -0.261088763722

1044 1051

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.323360674893 0.736314622780 0.000000000000 0.049409076176 0.310446284473 0.000000000000 0.641681674809 -0.200752320489 -0.143447847126 0.339700457654 -0.628330208939 0.185552173944 0.639301667753 0.365101155387 0.681054535772 0.396469223771 0.319603553402 0.498919194238 0.488951302441 0.379130726291 -0.307478690170 0.260027147177 -0.311797647655 -0.264439229466 0.346594381494 -0.335388236923 0.377500987095 0.794481832371 0.372414587639 0.231758116908 0.093691083172 0.618535273508 0.216039357148 -0.016088756293 0.190814133043 -0.325015262201 0.791760526519 -0.095711259443 0.165532501838 0.396550190684 -0.116543419028 0.532458022878 -0.116379869987 0.299489387437 0.027338909581 0.393816451236 0.110986781676 -0.441309164101 0.698080235015 0.390063083406 0.600445481659 0.907739460953 0.061031437454 0.415071360939

27 1053 1057

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.276696289187 0.690530044187 0.000000000000 0.171261006519 0.559635310433 0.000000000000 0.451929295519 -0.129456850950 -0.544898967971 0.221330891175 -0.409697710940 0.240596135309 -0.059707330161 0.228484127024 0.236193996758 0.946832562250 0.210156725822 -0.058448681556 0.689486128597 0.611408191327 -0.304248712816 0.440464566780 -0.125987853159 0.735653500273 0.663822845236 0.042622055329 0.517836275117 0.270241247687 0.736075086598 0.258294610615 0.368345820209 0.387923768824 -0.372927472652 0.794041777033 -0.070799044284 -0.014811856484 0.133716609227 0.090371109389 0.552497590882 0.388005154758 0.834544561505 0.109635790458 0.218995904543 -0.031244197766 -0.436409461806 0.473496873073 -0.032126562535 -0.381864289762 0.832934251017 0.543356538493 0.104805560730 0.835892278112 0.060778793421 0.545518136879

1054 1062

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.221183803503 0.627252207708 0.000000000000 0.124676953427 0.483538585986 0.000000000000 0.520072074242 -0.101264131820 -0.616384413054 0.631577671352 -0.366696275412 -0.141958946990 0.473743137298 0.527954695476 0.159461768207 0.612948715135 0.517207535368 0.325338619106 0.497640642336 -0.419203540785 -0.160413496130 0.063345944985 0.041445070104 -0.361383210815 -0.287482522153 0.083789162154 0.200944992593 0.839214730355 0.307858430972 0.210500998654 0.437316592413 0.119159358336 -0.487106744376 0.739881432097 -0.613776150889 -0.164629103279 0.548273456766 0.005907990559 0.500244612512 0.795497425539 0.356684540680 0.070132280668 0.080465783207 0.088631520853 -0.379705839556 -0.088380286250 0.013428934314 -0.247571636999 0.850788843275 0.517333187268 0.092329396777 0.809411973147 0.424422322448 -0.405854592107

1055 1064

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.199666949075 0.599555675143 0.000000000000 0.251048347121 0.662624646123 0.000000000000 0.268371087704 -0.390398687539 0.123572250690 0.630407466821 0.034864587088 0.679708736578 0.178671037992 0.576187196356 0.363708941213 0.842967773749 -0.031791224429 -0.295163110748 0.608529357965 0.328533991447 -0.504559286265 0.343447444812 0.315308739049 0.498562947998 0.049347909438 0.047585527107 0.275430500853 0.804666185458 -0.471528110644 0.088483632204 0.980330143272 0.219956911967 -0.046377183903 0.785658175010 0.516377058138 -0.065406175875 0.193407754679 0.077371167883 0.553974762804 0.001570041164 0.072133002392 0.368013963592 0.136966560724 0.081534709932 -0.444422481053 0.849610092629 -0.360007983508 0.061285680878 0.865713543322 0.489238100104 0.105764560776 0.734744707556 0.461614005304 -0.497054046182

1056 1065

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.999999999994 0.000000000073 -0.000000007842 1.000000000000 0.000000000000 0.000000000000 0.271802905621 0.685367778513 -0.000000005822 0.063224786261 0.349931706085 0.000000000000 0.896016993509 -0.048752654511 0.267252613068 0.685162684008 -0.322313422506 0.401596487076 0.291988109173 0.036843296874 -0.525099859006 0.000499294950 0.369628399698 0.630621722300 0.597998386570 0.509785978725 0.301146429828 0.438156017741 -0.158516562208 -0.097059177689 0.341767748426 -0.344996126576 -0.150177636121 0.342301884633 0.785382440869 0.218957684452 0.733245678519 0.544861944839 0.084124824280 0.618470427390 0.033875748015 -0.380182221236 0.305854993332 -0.168305130290 -0.471509880489 0.253783283300 0.009816902387 0.550637019478 0.961293419491 0.203252418628 0.186019934291 0.165007358441 0.090796587319 -0.442117293429 0.375380168224 0.924317636282 0.068750538638

28 1068 1073

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.347320929587 0.757634497000 0.000000000000 0.294030726755 0.708242462179 0.000000000000 0.276386325646 -0.198816193063 0.283143708109 -0.040292843672 -0.112361349038 0.463505269951 0.675222034860 0.660323221093 -0.037495565998 0.385063169336 0.161789012373 -0.398999130928 0.001265461952 -0.078330797540 -0.024351038979 0.621589001899 0.331686300054 0.557656733095 0.205100741712 0.887931942761 -0.181791489403 0.103279296536 -0.336085495014 0.023394738340 0.874894080168 0.175504785253 0.027551360974 0.570968175228 0.506827061175 0.289412661358 0.149468760102 -0.483176123425 -0.172192388741 0.788478965487 -0.327139892096 -0.217724575465 0.413839481288 0.193048749221 0.515432540558 0.072485772395 0.083270797899 0.347002626805 0.715688024165 0.608631637607 -0.342575804415 0.918457182255 0.384420587271 -0.093044163958

1070 1074

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.159287462737 0.541481698388 0.000000000000 0.133248925597 0.498740989914 0.000000000000 0.452980170800 -0.407397467797 -0.115641347325 0.335744002161 -0.478570205367 -0.062117401307 0.826205879375 0.500576208292 0.074848479145 0.619835157814 0.392683069636 0.338144945256 0.178660657168 -0.226415521254 -0.153555133716 0.216342481549 0.561083420138 -0.561207589711 0.632121283963 0.567901660268 0.250714176379 -0.123417681982 -0.311173387669 -0.209453282685 0.421142134367 -0.211656922267 -0.339011365573 0.187908041660 0.638344828993 -0.170833986660 0.430485352212 -0.395737114228 0.643855455114 0.636271222835 -0.115606020717 0.309299925899 0.572728317667 0.300201439049 -0.060017025412 -0.091677425042 -0.088230828608 -0.375785006728 0.725661396865 -0.624070869241 0.289743140151 0.399059444251 0.775701252373 -0.488916278130

1071 1075

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.320387658575 0.733571445316 0.000000000000 0.050315486428 0.313208117203 0.000000000000 0.061607418801 -0.228451970033 -0.086854682219 0.225290547043 -0.670010332096 -0.051625663387 0.073695281875 0.630171526178 -0.599318900795 0.169150861530 0.327689430601 -0.013631989927 0.337235356429 -0.025593900744 0.108154302908 0.021479422664 -0.656163252935 0.087496854827 0.190251955558 0.126427253942 -0.869232748569 0.460006847701 0.184606121615 -0.229994178604 0.462896515470 0.195970416569 0.090365444677 0.642885993679 -0.115692392653 0.706155392936 0.377459321184 -0.083872033296 -0.865871288086 0.150043869612 0.004541574585 -0.155616290207 -0.095281371072 0.578946955475 -0.285193041542 -0.061524555805 0.810839963562 0.396761940085 0.576145940286 0.060865960197 -0.815077413735 0.693051205175 0.677688900625 -0.245800286770

1072 1076

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.165249950050 0.550629053091 0.000000000000 0.269460969040 0.682870942597 0.000000000000 0.134341040454 -0.370863523257 0.387164139664 0.681493819513 -0.156850220478 0.353690964563 0.344216223103 0.515466930978 -0.025595758585 0.445649503757 0.174933158011 -0.559707770099 0.224923060434 -0.477390601099 -0.027345712500 0.284902902020 0.331702508253 0.414758186645 0.301793145935 0.377596312811 -0.540267116809 0.917817382758 -0.370766701000 0.089248939275 0.631438555737 0.025544685627 0.335742903450 0.439353448103 0.054455564261 -0.679033748518 0.141893325042 -0.127424166521 -0.522712685021 0.484110031835 0.319920463021 0.284047367349 0.255788774769 0.129110558425 0.437088211743 -0.197776501470 -0.392483252610 0.118180849777 0.966486350143 -0.148821355799 -0.209180159303 0.412934547688 0.325827276418 -0.215078806233 0.666061840025 -0.621153418738 -0.412952849188

29 1077 1083

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.999999999980 0.000000005093 -0.000000001684 0.052334747758 0.319265672588 0.000000000000 0.122715518548 0.479970773617 -0.000000001823 0.774227807783 -0.360666122829 0.128697954406 0.861747440976 -0.185421680459 0.105283898764 0.336638211828 0.490036612793 0.419937033372 -0.065080481830 -0.170257378812 -0.269896118642 0.414121017525 -0.280340969793 -0.212925349038 0.652297079341 -0.020256510557 0.410448774932 0.521765476270 0.585504450990 0.275668938495 0.484533122215 0.162803758130 -0.558232902765 0.125870818218 -0.184813704668 0.775546282239 0.311527446458 -0.396806783177 0.252263906767 0.511685180828 0.459330009343 0.115072094020 0.772866828445 0.423913000913 -0.084759847082 0.499764614909 -0.530978821989 0.253442317309 0.823814591027 -0.547995335277 0.145019419946 0.767175753726 0.400182219866 0.501293879674

1084 1078

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.170768676687 0.558905548762 0.000000000000 0.070288147928 0.368287757217 0.000000000000 0.423598764798 0.164295556110 0.883379816526 0.699058903187 -0.401401557222 -0.110569871884 0.647761944004 -0.088570920756 -0.057794669935 0.005747872289 0.243069583261 -0.433024153170 0.359461924174 0.840816057460 0.172689479817 0.700528902036 -0.147369180618 0.170993141586 0.181166107691 -0.114631632232 -0.062534485982 0.546131555945 0.559483440642 -0.519311741301 0.299931734920 0.625219376513 0.599670324215 0.684486011823 -0.425352188249 -0.414636870370 0.568765118194 0.362872389495 -0.327099696039 0.669143108953 0.522017619035 -0.094863444994 -0.029038735794 -0.008028033601 0.383578450737 0.003012542748 -0.153909529148 -0.410129299655 0.714958576275 0.648754304127 0.260676211203 0.953669347070 0.124761362721 -0.273768841965

1085 1080

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000060 0.000000002791 0.000000001638 1.000000000000 0.000000000000 0.000000000000 0.070688083933 0.369295768590 -0.000000002148 0.077354829353 0.385649956672 0.000000000000 0.276859498358 -0.356657570556 0.656113629188 0.244184999440 -0.511364422062 -0.409307828771 0.315375235740 0.023136229494 -0.268155383273 0.001804587929 0.149368247640 0.301103372791 0.036854896889 0.376746969923 0.624809266128 0.359734830005 0.554751622636 -0.540056695855 0.379430467193 -0.359146755126 0.040769537966 0.524085657887 0.045289575674 0.304595438505 0.314032040088 0.634023966641 -0.055848075322 0.436140339957 0.349938004253 -0.643800552639 -0.164618271214 -0.197823892065 0.225084335543 -0.279302715325 -0.058302576359 -0.076807728180 0.168109390351 0.518543228219 0.838362781475 0.523575210542 0.526120097062 -0.194439250086 0.120927309214 -0.191268381118 -0.762971077631 0.260963093274 0.779722443955 -0.569149518441

1090

1082 0.000000000000 0.000000000000 0.000000000000 0.999999999994 -0.000000000360 0.000000003321 0.337563715215 0.749118260415 -0.000000003672 0.000000000000 0.000000000000 0.000000000000 0.665676915476 -0.068142238483 0.473737270324 1.000000000000 0.000000000000 0.000000000000 0.897129492854 0.147535026059 -0.474900185187 0.077146569142 0.385151327602 0.000000000000 0.473191483963 0.424650244573 0.387354835249 0.286570519522 -0.579860030704 -0.157780503698 0.577171909229 -0.145193735227 -0.427792948780 0.293218234768 0.405844280066 -0.326133762090 0.690048756191 -0.017005224727 0.557512345619 -0.019530278575 0.116092535428 0.578428172705 -0.006452210629 0.561569222517 0.133085659098 0.568015326702 0.395371444654 -0.181041018627 0.949451260310 0.313042280070 -0.023384507929 -0.356275256745 0.060622595283 0.002344163970 0.542186776667 0.385067012819 0.298151186277 -0.018532061843 -0.131674695214 -0.348820522006 0.797501592192 0.443145714047 -0.409405772521

30 1091 10100

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000000 -0.000000003242 -0.000000008185 1.000000000000 0.000000000000 0.000000000000 0.005769707732 0.107266608324 -0.000000000125 0.240940477857 0.651021229949 0.000000000000 0.795084073560 -0.183704940982 -0.540664766327 -0.012780851898 -0.303057351908 -0.159246809923 0.433260594849 0.139341393286 0.333820703070 0.893604728975 0.119377906566 -0.162938853372 0.305630130519 0.199428011314 -0.656179328707 0.073907572706 0.276811459533 0.387798151382 0.643948149307 -0.291838559445 0.146440179303 0.141128105649 0.268416249824 -0.609904670306 0.232335253477 0.606122360365 0.302135428143 0.601831660991 -0.174512617149 0.159228772535 0.367197681911 -0.196291371259 -0.279194365276 -0.120384019016 0.489865555459 0.351598612463 0.341799321375 0.768532668589 -0.540861129356 0.304336208105 -0.167989233760 -0.270365859976 0.821315590484 -0.007364171067 0.570426568291

1094 10101

0.000000000000 0.000000000000 0.000000000000 1.000000000020 0.000000002720 -0.000000003949 0.000000000000 0.000000000000 0.000000000000 0.210135095219 0.613280878155 0.000000004339 1.000000000000 0.000000000000 0.000000000000 0.526748406461 -0.052196857784 0.675940381975 0.121037379708 0.476890671045 0.000000000000 0.455019031302 0.363491023189 -0.230734026154 0.673848032866 -0.075270303972 -0.624114283944 0.505124603514 -0.420826783510 0.387598416577 0.703035373024 0.266915297190 0.315064673195 0.246492577522 0.544974638969 0.405958359132 0.139531953501 -0.555579367930 0.237822055195 0.825736760075 -0.033194856768 -0.168668660053 0.628442824865 0.110511846542 -0.325461757696 0.175479987160 0.014593573085 0.589541334099 0.418304825606 0.404397596343 0.606993703616 0.925175949959 0.377795489491 0.036332213368 0.950249041541 0.029842231365 -0.152442685886 0.408727382618 0.078608384031 0.686828518953 0.702869838500 0.710682040006 -0.030083685623

1095 10103 0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.314323185864 0.727906111086 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.158546572160 -0.063852132939 -0.590637392689 1.000000000010 0.000000006065 0.000000003547 0.822703347529 0.177715976660 0.116851565639 0.018960290862 0.193806840555 0.000000000651 0.124138078041 0.440549449130 -0.548674639717 0.356936968390 -0.339698568259 0.775334602104 0.341600062394 0.340734857275 0.422277126652 0.540621665830 -0.116630990376 -0.182005874232 0.069596495056 -0.278582906643 -0.314241418023 -0.242304947670 0.394584837397 0.172513935286 0.700324158866 0.497352332484 -0.303891597472 0.500619216492 -0.272128441912 0.232155276040 0.229222278409 0.053735090583 0.458516928791 0.427964164870 0.566980004082 -0.306934938447 0.487161224639 0.871516389063 -0.055974322713 -0.083486660694 -0.039400806722 0.301933163297 0.106933714219 0.942176862553 0.317597135441

1097 10105

0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.386903099172 0.790007715275 0.000000000000 0.999999999997 0.000000008278 0.000000004627 0.600379257457 -0.177323892366 -0.136738768100 0.421474190912 0.815664086349 0.000000006316 0.306689800522 0.610562893525 0.404538347159 0.626988867636 -0.120351838092 0.285723486074 0.408929697235 0.315639640322 -0.545497061781 -0.057133112140 0.219817823101 -0.359459990455 -0.101905450829 -0.173632077250 0.161369713422 0.715577625773 0.325577919270 0.266425716019 0.321954778613 0.706387662422 -0.052891257555 0.144478816059 -0.406796214850 -0.104349522174 0.738070699603 -0.153173441689 0.243760471915 0.988889566369 0.100468515784 -0.276549921967 0.301994825091 0.281530481696 -0.544192782194 0.394694417825 -0.065223071327 0.510519594232 0.724738681049 0.622447505886 0.295487642055 0.809926204303 0.569219987375 -0.141450166410

31 10106 10112

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000010 -0.000000009907 0.000000003092 0.999999999966 -0.000000000404 0.000000004688 0.311960823194 0.725673533203 -0.000000006234 0.120686625974 0.476243624489 0.000000003585 0.141613105169 -0.205341943516 -0.322787609860 0.289671968935 -0.437900366712 -0.368489775394 0.114163390974 0.396089272152 0.475665272891 -0.071299007324 0.356817035490 0.119489942355 0.514897953039 -0.206195163554 -0.214744220993 0.325888083381 -0.504291108191 -0.197898116442 0.701365250085 0.562190250771 0.397476165920 0.479283618267 0.431160408037 0.120535975141 0.197666883183 0.592350574934 -0.465876780466 -0.130256435610 0.464605614139 -0.671513480943 0.141122674381 0.244419753037 0.469936713561 0.217426779687 -0.118391155680 0.062810118138 0.577640075963 0.774859214651 -0.256758914375 0.596568037160 0.801823272078 -0.034438603338

10107 10115

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.999999999982 -0.000000000169 0.000000001243 0.999999999916 0.000000005131 -0.000000000424 0.161582223391 0.545028101742 -0.000000003523 0.081674068459 0.395825071912 -0.000000001027 0.536689289970 -0.300816543584 -0.379264455594 0.417444228134 -0.375029087066 0.541333782741 0.799724066235 0.595106254752 -0.021304896544 0.678113931348 0.323625515236 -0.124949229576 0.449424539419 -0.322918228887 -0.207103951880 0.696822248746 -0.534171656989 0.388698519562 1.172455809590 0.171181235858 0.275691554865 0.487648758352 0.230935403748 -0.220282298306 0.478520666504 -0.184875392102 -0.350150034424 0.330700175882 -0.113254077899 0.705407035744 0.083050584948 0.331648379612 0.409327817660 0.627103979155 -0.275742581909 -0.235731780508 0.880895030917 0.435890296809 -0.184454855307 0.626180031598 -0.176822391277 0.759363160787

10110 10117

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000070 0.000000001469 0.000000007495 1.000000000020 -0.000000003843 0.000000004361 0.188066191724 0.583749511944 0.000000000199 0.116116020664 0.467706219187 -0.000000003107 0.320086189720 0.053853975023 -0.837723963939 0.679713502925 -0.343063182976 -0.158148207358 0.767090139228 0.231523172470 0.038986436024 0.880435077212 0.567939292132 0.202109117292 -0.020893217598 -0.320405146806 -0.233882323174 0.582521295641 -0.252722845519 -0.285498342215 0.211759055663 0.652144152024 -0.229323691927 0.513747125258 0.346709020691 0.511967363169 0.604317366357 -0.185684855770 0.150071255880 0.745911626556 -0.056318073967 -0.373283338771 -0.172058915620 0.243552992519 -0.311442274114 0.363890486581 -0.191378675405 0.540947749372 0.758665127764 0.093117328132 -0.644791739339 0.929849440697 -0.236008091765 -0.282276811395

10111 10118

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000010 0.000000001386 -0.000000003394 1.000000000010 -0.000000005221 -0.000000003477 0.575666494042 0.905505979154 -0.000000007830 0.403387024657 0.802529095595 0.000000003514 -0.046761271941 0.249209720978 0.426449165728 0.291456495118 0.049521625062 0.648422169921 0.754043104366 -0.094650752072 -0.063930611537 0.772966779876 0.410705206294 -0.150135446229 0.350433910329 0.765036052358 0.249179938929 0.368766466185 0.447188590384 0.763807149946 -0.171099357115 0.212454003569 -0.400940150249 0.058350028966 -0.188073168195 0.056640879831 0.403012464720 -0.109098114248 0.352053775060 0.780978266324 0.492948233281 0.175039702823 0.056772290999 0.813684852832 0.182969956306 -0.169989630179 0.292172483134 0.410303720418 0.930681038358 0.365784224418 -0.005891178773 0.736076957506 -0.037246157127 0.675872352178

32 10119 10137

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.999999999996 0.000000000741 -0.000000006296 1.000000000130 0.000000004781 0.000000001563 0.340261376595 0.751495142976 -0.000000004954 0.249377129838 0.660730893110 -0.000000000056 0.600057360268 -0.211491712042 0.071849587510 0.176113362065 -0.325312806405 0.149499976397 -0.005977751685 0.389656556064 -0.449055588915 0.141315999593 0.624954976615 0.458983875969 0.915178276694 0.048257887855 -0.635920568908 0.733727173223 0.032665287622 -0.087131428393 0.711114312810 0.524466897641 0.219405613118 -0.143269035537 -0.229286273183 0.315682461120 0.176993010548 -0.003999403094 -0.440471499252 0.414900840467 0.357311464511 -0.271130527509 0.490434990177 0.054148731917 0.507353864303 0.140599785814 0.576837306676 0.665121220719 0.891729311084 0.225405057919 -0.392442856525 0.770962015821 0.627391661111 -0.109532067039

10126 10138

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000010 0.000000003976 0.000000007291 0.999999999967 0.000000000117 0.000000003265 0.217612139530 0.622791490605 0.000000001368 0.087149291174 0.408293501663 0.000000002913 0.327086697642 -0.283246277787 -0.408791979161 0.609678949988 0.122983224062 0.803467986810 0.750426751072 0.347174937859 0.241863298244 0.438081611799 0.346487167790 -0.156011216013 0.337899755351 0.646705845240 -0.618428927296 -0.114571777741 -0.185174892401 0.485790688589 0.297989001628 0.170846938898 0.260186707121 0.170414516351 0.768442059157 0.388849872056 0.677526694544 0.406973043112 -0.634349822374 0.249034133238 -0.110032975235 -0.082425599242 0.493815984594 -0.208204533359 0.132335852769 0.453267635638 0.834524392164 0.174673712765 0.670682334898 0.696085786024 -0.256222138284 0.989387164351 0.049826332784 -0.136493134228

10131 10142

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.999999999973 0.000000004142 0.000000006489 1.000000000010 -0.000000001127 -0.000000002355 0.162315445140 0.546154365618 0.000000002933 0.094000941487 0.423279700711 0.000000001667 0.398595003167 -0.419849857781 -0.104918111887 0.681561997645 -0.276319429726 -0.406611682869 0.922833634587 0.349517589211 -0.469949054182 0.840804415986 0.345100417609 0.360512313327 0.404736530806 0.035370365684 0.325592844786 0.132697870703 -0.200282904394 -0.087976429665 0.594365756366 -0.268966431969 -0.607905880677 0.271922389301 0.750038497312 0.190422145893 0.852133546012 0.060424972235 0.300420640739 0.532584662519 0.164854373645 -0.577441592493 0.034555092237 0.124060283644 -0.271869512119 0.448052981231 -0.068588826169 0.391247556472 0.858185225071 -0.442931685725 -0.259479558478 0.635177143830 0.760312740973 -0.135921050088

10133 10143

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000040 -0.000000006245 0.000000005346 0.999999999991 -0.000000002855 -0.000000004288 0.169304635538 0.556727226301 0.000000002465 0.109027988130 0.454058224802 0.000000002271 0.309216115424 -0.428863148029 -0.095059935559 0.278056935528 -0.491086608122 -0.279518261470 0.699743797394 0.453630770776 0.167032820713 0.420527786752 0.405890531196 0.138971963151 0.798261523519 -0.060410270682 -0.685056414833 0.542435971636 -0.091848902160 -0.719744434143 0.513789648487 0.086764080223 0.262263733136 0.274105755269 -0.164988748227 0.240802003024 -0.102374999472 0.363461273273 -0.475150526258 0.447437330387 0.485444216805 -0.498720223189 0.723991384671 0.300339693632 0.084433484708 0.423182571126 -0.240788863380 0.188300314991 0.721681496369 -0.616524164284 -0.314759864997 0.766058087681 0.599855731874 -0.230928792572

33 10147 10164

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 1.000000000060 -0.000000004507 -0.000000001893 0.999999999915 0.000000008767 0.000000003791 0.168148736299 0.554998621664 -0.000000000317 0.440968420486 0.829146372870 -0.000000004522 0.675384280182 0.138470215813 -0.754464174494 0.261434704851 -0.131514119771 -0.211893994109 0.295777207594 0.014574093139 0.162349994884 -0.227952932593 0.477723210189 0.412068838213 0.992100092548 -0.217983496853 -0.516657668719 0.316467765015 0.686665829179 -0.400303666246 0.247705872001 0.446740465402 -0.453259731467 0.483712303642 -0.112304541499 0.177342948132 1.163565579370 0.142154311123 -0.191671751639 -0.176052053917 0.142765091667 -0.529516638157 0.404702153809 -0.245344706677 -0.715093964382 0.434803016702 0.587258201604 0.125679483542 0.726283256480 0.611959112962 -0.313079343671 0.477682108892 0.291714846266 -0.828687064779

10148 11n71

0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.999999999952 0.000000007534 -0.000000004692 1.000000000000 0.000000000000 0.000000000000 0.098970865365 0.433758579272 0.000000003063 0.065457877472 0.355852527348 0.000000000000 0.369674821817 -0.509648691004 -0.191577893303 0.357888038312 -0.560829036635 -0.272359158619 -0.333232775899 0.187574631815 -0.332292310433 0.513047120384 0.334296663791 0.145582990113 0.443646749824 -0.155361614820 0.195772963204 -0.038318699501 -0.126403727709 -0.549939031059 0.356665365760 0.588958014922 -0.466361862645 0.648478789572 -0.181945510740 0.174784584771 -0.001569375787 -0.344634792001 -0.474867203670 0.135447038614 0.464800325715 -0.389589622239 0.480086430748 0.011300956996 0.325955764673 0.250854179559 -0.518112727856 -0.246191153148 0.745953423255 -0.344430657134 -0.570018431930 0.208000734576 0.399875823250 0.148093772117 0.757283536090 0.124949336453 -0.641022081747

10149 11n73

0.000000000000 0.000000000000 0.000000000000 1.000000000030 -0.000000007092 -0.000000000798 0.000000000000 0.000000000000 0.000000000000 0.252061932081 0.663768511711 -0.000000003399 1.000000000000 0.000000000000 0.000000000000 0.665548785166 -0.228702690211 -0.180343498020 0.143944996500 0.516884736650 0.000000000000 0.005253988030 0.183105071809 0.447689262271 0.254278142296 -0.476425971543 -0.034065142393 0.389275547414 -0.086322312011 -0.435450775737 0.345559543009 0.495701711281 0.181887338885 0.798392082260 -0.056610644564 0.476547512489 0.142457394856 -0.317577470097 -0.363389187799 0.605524514965 0.331978506235 -0.424452250762 0.041994233470 0.374785753440 0.351131854507 0.302924972937 -0.351415042519 0.239932257401 0.593243691728 0.136260749026 -0.448386661251 0.935296830455 0.328867546275 -0.130636809313 0.311183433545 -0.338160018004 0.385499989658 0.047392356781 0.524930993525 -0.045183390143 0.926163667162 0.153273809558 -0.344569297721

10153 11n74 0.000000000000 0.000000000000 0.000000000000 0.999999999959 -0.000000004279 0.000000006156 0.061021249554 0.343975149211 0.000000002695 0.000000000000 0.000000000000 0.000000000000 0.585049572528 -0.507000959960 0.035128031064 1.000000000000 0.000000000000 0.000000000000 0.388272275902 0.456561328269 -0.146052575620 0.187258369934 0.582624272372 0.000000000000 0.033013225049 -0.175236990370 0.542875498523 0.373601037342 -0.397958004908 0.061113073781 0.625938169312 -0.137767576002 -0.261510014435 0.332844653560 0.460274803417 -0.450526803374 0.502915798439 0.444325282470 0.542252015246 0.203121821277 0.315634308944 0.530417187965 0.217912089502 0.145156597949 -0.368391155314 0.618743828278 0.101637083703 -0.353587073838 0.955714478091 0.014816482956 0.293922282366 -0.343427650530 0.163126483758 -0.088172117500 0.364323742898 0.350568531364 0.592969162918 0.294774938190 -0.097657189661 -0.298241624032 0.820482069206 0.449750181243 0.352893678866

34 11n75

0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.128736651836 0.490815829151 0.000000000000 0.631215510645 -0.319092005824 0.302595927427 0.423354521122 0.658741434549 0.327806479237 0.256871440448 -0.140387371664 -0.249840934787 0.504176582189 -0.123995706072 0.718958053738 0.310512766975 0.177727523438 -0.214560783485 0.301484024421 -0.163999202678 0.725195209561 0.742187966034 -0.029424262308 -0.162312323517 0.567379979547 0.332135799239 0.753502335546

11n76

0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.042804337102 0.289441639935 0.000000000000 0.591760594469 -0.341964035913 -0.547698731060 0.362088134700 -0.167403562763 0.409787144783 0.154310730233 0.327853765638 -0.433746333425 0.133910295057 -0.325253890459 0.323243898871 0.957595852805 0.090809240202 -0.062026880271 0.027836273921 0.319229693056 0.226713872293 0.782414764654 -0.242767173015 -0.112064556551 0.684558204841 0.690938040821 0.232346052110

11n78

0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.045703204137 0.298860545081 0.000000000000 0.527323557198 -0.480701271458 0.400406305785 0.396499801529 0.336626928291 -0.160717351574 -0.016622961984 -0.230283084764 0.551983581051 0.267723061918 0.258479373949 -0.272794327021 0.605351188579 0.084995994287 0.652360127688 0.067489613226 0.529951401016 -0.063684292569 0.414814083497 -0.363693193181 0.220507634856 0.770144644100 0.568516256912 0.289251608103

11n81

0.000000000000 0.000000000000 0.000000000000 1.000000000000 0.000000000000 0.000000000000 0.194328448724 0.592362516931 0.000000000000 -0.058436535389 -0.372011891103 -0.078050393573 0.490713559007 0.456389837504 0.032334160902 0.323902369454 -0.208341913181 -0.695886771694 0.043662106274 0.155002350414 0.192621157930 0.505318611528 0.200236875514 -0.693283525936 0.428023583539 -0.209659289626 0.215567733871 -0.159524316595 0.434611591425 -0.274026482677 0.835813573457 0.507097141963 -0.210399997713